User peter - MathOverflow

Answer by Peter for Which functions of one variable are derivatives ?

2011-05-30T21:46:51Z

Take a look a this book by Andrew M. Bruckner: Differentiation of real functions.

Chapter seven is about The problem of characterizing derivatives.

There is a review by Daniel Waterman.

You might also want to take a look at Homeomorphisms in Analysis by Goffman, Nishiura and Waterman.

Generalized Gauss-Green theorem

2011-06-07T21:12:53Z

I am looking for a generalized version of the Gauss-Green theorem also known as the divergence theorem:

http://en.wikipedia.org/wiki/Divergence_theorem

A quick search on MathSciNet suggests that there are generalizations for bad domains and nonsmooth functions. However, they seem to rely on heavy machinery and not to be suited for the special case I am interested in.

For example, I found this formula on PlanetMath:$$ \int_E \mathrm{div} f(x)\, dx = \int_{\partial^* E} \langle \nu_E(x),f(x)\rangle \,d\mathcal H^{n-1}(x)$$

See http://planetmath.org/?method=l2h&from=objects&name=GaussGreenTheorem&op=getobj for the details.

Let $\Omega \subset \mathbb{R}^n$ be open and bounded and $f\in C^1(\Omega, \mathbb{R}^n) \cap C^0(\overline\Omega, \mathbb{R}^n)$.

Question: What conditions do we have to impose on $\Omega$ (or $f$) to ensure that the divergence theorem holds true?

To clarify my question: I know that requiring the boundary of $\Omega$ to be piecewise regular is sufficient for the Gauss-Green theorem to be true. I wondered if this condition is also necessary. If so: is the an other "version" of Gauss-Green (e.g. the one cited above) which holds true under weaker conditions and is especially suited for the case of an open and bounded domain

Topological spaces, uncountable subsets and separability

2011-05-27T22:32:11Z

Hi, the following is a well known theorem

Let $M$ be a metric space. If every uncountable subset of $M$ has a limit point, then $M$ is separable.

Question: Is there a similar result for topological spaces?

I have almost no knowledge of topology so I can only hope that this is not trivial.

Comment by Peter

2011-06-07T22:09:09Z

Wikipedia says that the divergence theorem is also known as Gauss' theorem, Ostrogradsky's theorem, or Gauss–Ostrogradsky theorem. My professor called it the Gauss-Green theorem.